The parents of Jeremy Forrest, the maths teacher who has absconded with 15-year-old Megan Stammers, struggled to hold back tears at a press conference on Thursday as police confirmed they have issued a European arrest warrant for Forrest for child abduction.

Looking exhausted and tense, Forrest's parents appealed for him to get in touch. "Hi Megan, hi Jeremy," said Jim Forrest, "I hope this message reaches you and you are both okay. There are a lot of people back home that are desperate to hear from you."

He said he was worried that his "caring" but "vulnerable" 30-year-old son could "flip". Clasping the hand of his wife, Julie, and struggling to steady his voice, he said: "All I am asking is for one of you to make a call, send an email, so we know you are both safe. We are all here for you both. Please, please get in contact."

Despite desperate pleas by Megan's parents, there have been no confirmed sightings of Megan, Forrest or Forrest's black Ford Fiesta car – registration GJ08 RJO – since CCTV footage showed the pair, arm-in-arm and holding hands, on board a Dover-to-Calais ferry at 9.30pm last Thursday.

Forrest's parents sat with their heads in their hands next to Chief Inspector Jason Tingley as he said the warrant meant their son could now be arrested and Stammers taken into care.

"This has probably not gone the way that Megan or Jeremy expected," said Tingley. "Jeremy will be aware that he has questions to answer, and making contact with us is the best way he can do this.

"As part of the process of working with the French authorities we have secured an international letter of request and a European arrest warrant," he added. "These orders, which have been obtained with the help of the Crown Prosecution Service, enable us to work with the European authorities, and are in relation to an offence of child abduction. This means taking Megan without the consent of her parents.

"The full details of this warrant have been circulated to every EU country," he added. "When the pair are found in an EU country Jeremy can be arrested and Megan will be taken into protection."

Sussex Police had previously said it was working with Interpol. Despite his assurances, however, neither Megan nor Forrest have a profile on the website of the international crime agency under its "yellow notices" section which "help locate missing persons, often minors, or to help identify persons who are unable to identify themselves".

Forrest is not listed either, under the website's "red notices" – profiles used "to seek the location and arrest of wanted persons with a view to extradition or similar lawful action".

Tingley defended his force against accusations that they had not acted swiftly enough when Megan was reported missing because they were expecting the couple to use their return tickets to come home on Sunday evening.

But he acknowledged that his officers had not received the warrant as quickly as they would have liked. "The warrant was issued on Tuesday afternoon although we applied for the warrant and the letter of request on Monday morning," he said. "It didn't get done as quickly as we might have liked. It could have been applied for on Friday but … we believed they would return on Sunday."

He said an officer from the Sussex force had gone to Paris to work with French police searching for the pair. But he refused to comment on whether Forrest was being investigated before the pair set off to France.

"That is part of the investigation which I'm not willing to speak about," he said. "There was clearly a line of inquiry in regards to that and we'll be following that through."

Tingley also refused to comment on suggestions that the pair had driven across Spain's open border with France, or on reports that UK officers had interviewed the couple and taken their phone just before they fled the country.

Tingley said the French authorities had "a slightly different perspective", because what had happened was not seen as an offence in that country, "but we have their full support, and we have that team working with us.

"There have been rumoured sightings and information passed to us, from countries throughout Europe, and we are in liaison with those specific countries and the French authorities and Interpol. That is being done very quickly."

Tingley said the tickets had been bought in advance. "We understand that Megan went willingly with Jeremy, so there was an element of pre-planning."

Megan's stepfather, Martin Stammers, made a fresh appeal for people to help find the teenager. Thanking people for their support in spreading appeals online, he tweeted: "Please continue your efforts, if I could thank you all individually I would, your help is invaluable. I implore you to RT #findmeganstammers."

He added: "She is by no means in any trouble whatsoever. We just want to hug her. We want her back in our arms with her smiley face and dimples."

The case is set to feature on the Crimewatch TV programme on Thursday night.